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I need a help to find a counter example (or to prove) the following

Let $G$ be a compact abelian topological group. Assume there exists a dense subgroup $H\leq G$ such that every element of $H$ is of finite order then $G$ is a Lie group (equivalently, is it isomorphic to a direct product of a torus with a direct product of finite groups).

Note that the converse is true, every compact (abelian or non-abelian) Lie groups contains a dense subgroup of torsion elements

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    $\begingroup$ Hmm, your notion of a Lie group looks too wide for me. The subgroup $t(G)$ of all periodic elements is dense of any compact abelian topological group $G$ of the form $\Bbb T^\kappa\times \prod A_\alpha$, where $\{A_\alpha\}$ is a family of finite abelian group whose periods are bounded in common. Or you are asking whether such form of $G$ is necessary to have the subgroup $t(G)$ dense? $\endgroup$ – Alex Ravsky Jun 1 '17 at 3:42
  • $\begingroup$ yes exactly, i asked whether such form is necessary to have the subgroup t(G) dense. $\endgroup$ – Yanko Jun 1 '17 at 17:13
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    $\begingroup$ Just for general knowledge, if G is countably based profinite group (so no $S^1$ parts) and t(G) is dense in G then G is a direct product. $\endgroup$ – Yanko Jun 17 '17 at 22:41
  • $\begingroup$ @Yanko what do you mean by "is a direct product"? $\endgroup$ – YCor May 6 at 23:16
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There is a lot of counterexamples, but some efforts are needed in order to prove that a candidate group is indeed a counterexample. Also I may ask my teacher about them. The first family consists of zero dimensional candidates, which are closed subgroups of the product $\prod A_\alpha$ of a family $\{A_\alpha\}$ of finite abelian group whose periods are bounded in common. Candidates of bigger dimensions constitute the second family is so-called exotic tori, introduced by Dikranjan and Prodanov, I guess in [DP], that is compact abelian groups $G$ such that torsion part $t(G)$ has-non zero intersection with each non-zero closed subgroup of $G$. For us is essential that by Proposition 3 from [DP], torsion part of an exotic torus is dense in it. From the other hand, the motivation for introducing the exotic tori in [DP] was the fact that they are precisely the completions of the torsion minimal abelian groups. The structure of exotic tori has been described in detail in [DP]. In particular, Proposition 2.5* describes the structure of $n$-dimensional exotic tori as follows. A compact abelian group $G$ of dimensional $n$ is an exotic torus if and only if there is a representation of $G$ as a projective limit $$G=\lim_{\longleftarrow} G_m$$ such that $G_0=\Bbb T^n$, the continuous homomorphisms $\sigma_m$ in the projective system $$G_0\stackrel{\sigma_1}\longleftarrow G_1\stackrel{\sigma_2}\longleftarrow G_2\cdots \stackrel{\sigma_m} \longleftarrow G_m\longleftarrow\cdots$$ are epimorphisms and $\ker\sigma_m $ is a compact $p_m$-group for each natural $m$. Thus, according to Example 1* of [DP], for each natural $n$ there is a continuum non-isomorphic connected $n$-dimensional exotic tori (so here we have continuum many connected counterexamples). For more details on exotic tore see [DP] or [DPS]. See also [CD], in particular, Theorem 4.1 and Remark 4.2(c). Also you can google for exotic torus Prodanov Dikranjan.

References

[CD] W. W. Comfort, Dikran Dikranjan, On the poset of totally dense subgroups of compact groups, Topology Proceedings, 24, Summer 1999, 103–127.

[DP] D. Dikranjan, I. Prodanov, A class of compact Abelian groups, Annuaire Univ. Sofia, Fac. Math. Méc. 70 (1975/76), 191–206.

[DPS] D. Dikranjan, I. Prodanov, L. Stoyanov, Topological groups. Characters, dualities and minimal group topologies, Marcel Dekker, Inc., New York, 1990.

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  • $\begingroup$ I might have a use of the article of DP, if you still have it then it would be nice if you can scan it for me. $\endgroup$ – Yanko Jul 26 '17 at 15:03
  • $\begingroup$ @yanko OK, if this is urgent then I can do this. If not then I can assign this task to the student and I hope it will be done in a pair of weeks from now (such a long delay is because there are vacations now, and it is not so easy for me to catch the student). $\endgroup$ – Alex Ravsky Jul 27 '17 at 20:55
  • $\begingroup$ That's fine, I'll wait. Thanks $\endgroup$ – Yanko Jul 30 '17 at 16:28
  • $\begingroup$ @yanko Yesterday I gave the paper to the student. He said that he'll start to scan it next week. $\endgroup$ – Alex Ravsky Jul 31 '17 at 22:09
  • $\begingroup$ @yanko At last I received the link to the scanned paper (added to the references). $\endgroup$ – Alex Ravsky Oct 31 '17 at 8:11

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